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Representation Theory of Groups MAT 445/1196 - INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 – Representation Theory of Groups - Algebraic Foundations 1.1 Basic definitions, Schur’s Lemma 1.2 Tensor products 1.3 Unitary representations 1.4 Characters of finite-dimensional representations CHAPTER 2 – Representations of Finite Groups 2.1 Unitarity, complete reducibility, orthogonality relations 2.2 Character values as algebraic integers, degree of an irreducible representation divides the order of the group 2.3 Decomposition of finite-dimensional representations 2.4 Induced Representations, Frobenius reciprocity, and Frobenius character formula CHAPTER 3 – Representations of SL2(Fq) CHAPTER 4 – Representations of Finite Groups of Lie Type CHAPTER 5 – Topological Groups, Representations, and Haar Measure 5.1 Topological spaces 5.2 Topological groups 5.3 General linear groups and matrix groups 5.4 Matrix Lie groups 5.5 Finite-dimensional representations of topological groups and matrix Lie groups 5.6 Groups of t.d. type 5.7 Haar measure on locally compact groups 5.8 Discrete series representations 5.9 Parabolic subgroups and representations of reductive groups CHAPTER 6 – Representations of Compact Groups 6.1 Examples of compact groups 6.2 Finite-dimensional representations of compact groups 6.3 The Peter-Weyl Theorem 6.4 Weyl’s character formula 1 CHAPTER 1 Representation Theory of Groups - Algebraic Foundations 1.1. Basic definitions, Schur’s Lemma We assume that the reader is familiar with the fundamental concepts of abstract group theory and linear algebra. A representation of a group G is a homomorphism from G to the group GL(V ) of invertible linear operators on V , where V is a nonzero complex vector space. We refer to V as the representation space of π. If V is finite-dimensional, we say that π is finite-dimensional, and the degree of π is the dimension of V . Otherwise, we say that π is infinite-dimensional. If π is one-dimensional, then V ' C and we view π as a homomorphism from G to the multiplicative group of nonzero complex numbers. In the above definition, G is not necessarily finite. The notation (π, V ) will often be used when referring to a representation. Examples: (1) If G is a group, we can define a one-dimensional representation of G by π(g) = 1, g ∈ G. This representation is called the trivial representation of G. (2) Let G = R and z ∈ C. The function t 7→ ezt defines a one-dimensional representation of G. If n is a positive integer and C is the field of complex numbers, let GLn(C) denote the group of invertible n × n matrices with entries in C. If (π, V ) is a finite-dimensional representation of G, then, via a choice of ordered basis β for V , the operator π(g) ∈ GL(V ) is identified with the element [π(g)]β of GLn(C), where n is the degree of π. Hence we may view a finite-dimensional representation of G as a homomorphism from G to the group GLn(C). Examples: (1) The self-representation of GLn(C) is the n-dimensional representation defined by π(g) = g. (2) The function g 7→ det g is a one-dimensional representation of GLn(C). (3) Let V be a space of functions from G to some complex vector space. Suppose that V has the property that whenever f ∈ V , the function g0 7→ f(g0g) also belongs to V for all g ∈ G. Then we may define a representation (π, V ) by (π(g)f)(g0) = f(gg0), f ∈ V , g, g0 ∈ G. For example, if G is a finite group, we may take V to be the space of all complex-valued functions on G. In this case, the resulting representation is called the right regular representation of G. Let (π, V ) be a representation of G. A subspace W of V is stable under the action of G, or G-invariant, if π(g)w ∈ W for all g ∈ G and w ∈ W . In this case, denoting the 2 restriction of π(g) to W by π |W (g), (π |W ,W ) is a representation of G, and we call it a subrepresentation of π (or a subrepresentation of V ). 0 If W ⊂ W are subrepresentations of π, then each π |W (g), g ∈ G, induces an in- 0 0 vertible linear operator πW/W 0 (g) on the quotient space W/W , and (πW/W 0 , W/W ) is a representation of G, called a subquotient of π. In the special case W = V , it is called a quotient of π. A representation (π, V ) of G is finitely-generated if there exist finitely many vectors v1, . , vm ∈ V such that V = Span{ π(g)vj | 1 ≤ j ≤ m, g ∈ G }. A representation (π, V ) of G is irreducible if {0} and V are the only G-invariant subspaces of V . If π is not irreducible, we say that π is reducible. Suppose that (πj,Vj), 1 ≤ j ≤ `, are representations of a group G. Recall that an element of the direct sum V = V1 ⊕ · · · ⊕ V` can be represented uniquely in the form v1 + v2 + ··· + v`, where vj ∈ Vj. Set π(g)(v1 + ··· + v`) = π1(g)v1 + ··· + π`(g)v`, g ∈ G, vj ∈ Vj, 1 ≤ j ≤ `. This defines a representation of G, called the direct sum of the representations π1, . , π`, sometimes denoted by π1 ⊕ · · · ⊕ π`. We may define infinite direct sums similarly. We say that a representation π is completely reducible (or semisimple) if π is (equivalent to) a direct sum of irreducible representations. Lemma. Suppose that (π, V ) is a representation of G. (1) If π is finitely-generated, then π has an irreducible quotient. (2) π has an irreducible subquotient. Proof. For (1), consider all proper G-invariant subspaces W of V . This set is nonempty and closed under unions of chains (uses finitely-generated). By Zorn’s Lemma, there is a maximal such W . By maximality of W , πV/W is irreducible. Part (2) follows from part (1) since of v is a nonzero vector in V , part (1) says that if W = Span{ π(g)v | g ∈ G }, then π |W has an irreducible quotient. qed Lemma. Let (π, V ) be a finite-dimensional representation of G. Then there exists an irreducible subrepresentation of π. Proof. If V is reducible, there exists a nonzero G-invariant proper subspace W1 of V . If π |W1 is irreducible, the proof is complete. Otherwise, there exists a nonzero G-invariant subspace W2 of W1. Note that dim(W2) < dim(W1) < dim(V ). Since dim(V ) < ∞, this process must eventually stop, that is there exist nonzero subspaces Wk ( Wk−1 ( ··· ( W1 ( V , where π |Wk is irreducible. qed Lemma. Let (π, V ) be a representation of G. Assume that there exists an irreducible subrepresentation of π. The following are equivalent: 3 (1) (π, V ) is completely reducible. (2) For every G-invariant subspace W ⊂ V , there exists a G-invariant subspace W 0 such that W ⊕ W 0 = V . Proof. Assume that π is completely reducible. Without loss of generality, π is reducible. Let W be a proper nonzero G-invariant subspace of V . Consider the set of G-invariant subspaces U of V such that U ∩W = {0}. This set is nonempty and closed under unions of chains, so Zorn’s Lemma implies existence of a maximal such U. Suppose that W ⊕U 6= V . Since π is completely reducible, there exists some irreducible subrepresentation U 0 such that U 0 6⊂ W ⊕U. By irreduciblity of U 0, U 0 ∩(W ⊕U) = {0}. This contradicts maximality of U. Suppose that (2) holds. Consider the partially ordered set of direct sums of families of P irreducible subrepresentations: α Wα = ⊕αWα. Zorn’s Lemma applies. Let W = ⊕αWα be the direct sum for a maximal family. By (2), there exists a subrepresentation U such that V = W ⊕ U. If U 6= {0}, according to a lemma above, there exists an irreducible subquotient: U ⊃ U1 ⊃ U2 such that πU1/U2 is is irreducible. By (2), W ⊕ U2 has a G-invariant complement U3: V = W ⊕ U2 ⊕ U3. Now U3 ' V/(W ⊕ U2) = (W ⊕ U)/(W ⊕ U2) ' U/U2 ⊃ U1/U2. Identifying πU1/U2 with an irreducible subrepresentation π |U4 of π |U3 , we have W ⊕ U4 contradicting maximality of the family Wα. qed Lemma. Subrepresentations and quotient representations of completely reducible repre- sentations are completely reducible. Proof. Let (π, V ) be a completely reducible representation of G. Suppose that W is a proper nonzero G-invariant subspace of W . Then, according to the above lemma, there exists a G-invariant subspace U of V such that V = W ⊕ U. It follows that the subrep- resentation π |W is equivalent to the quotient representation πV/U . Therefore it suffices to prove that any quotient representation of π is completely reducible. Let πV/U be an arbitrary quotient representation of π. We know that π = ⊕α∈I πα, where I is some indexing set, and each πα is irreducible. Let pr : V → V/U be the canonical map. Then V/U = pr(V ) = ⊕α∈I pr(Vα). Because pr(Vα) is isomorphic to a quotient of Vα (pr(Vα) ' Vα/ker (pr | Vα)) and πα is irreducible, we have that pr(Vα) is either 0 or irreducible. Hence πV/U is completely reducible. qed Exercises: (1) Show that the self-representation of GLn(C) is irreducible. 1 t (2) Verify that π : t 7→ defines a representation of , with space 2, that is a 0 1 R C two-dimensional representation of R. Show that there is exactly one one-dimensional 4 subrepresentation, hence π is not completely reducible.
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